I am a bit confused as to what I should use to derive the equations of motions from the lagrange equation. Suppose I have a lagrange function: $$L(x(t), \dot{x}(t)) = \frac{1}{2}m\dot{x}^2-\frac{1}{2}k(\sqrt{x^2+a^2}-a)$$ Method 1: Principle of least action $$\delta L = \delta \dot{x}(m\dot{x...